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The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?

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    $\begingroup$ I don't think that this is a research level question, but $\langle x,y,z \mid yz=zy, y^x=z, z^x=yz \rangle$ is such an example. $\endgroup$
    – Derek Holt
    Commented Dec 2, 2014 at 1:54
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    $\begingroup$ Note that Derek's example contains RW's example with index two (because the square of the matrix $\begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix}$ is $\begin{pmatrix}2 & 1 \\ 1 & 1\end{pmatrix}$) $\endgroup$
    – YCor
    Commented Dec 2, 2014 at 13:25

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Semi-direct product of $\mathbb Z$ and $\mathbb Z^2$ determined by the matrix $\left( \begin{array}{ccc} 2 & 1 \\ 1 & 1 \end{array} \right)$.

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